% Statistical Mechanics Perspective and Coordination Mechanisms
% Sections 3-4 of Hierarchical Cooperation Paper

\section{Statistical Mechanics Perspective}
The system admits a statistical physics interpretation through a hierarchical Hamiltonian capturing multi-scale interactions. For binary states $s_i \in \{-1,+1\}$, the energy function is
\begin{equation}
    \mathcal{H}(s) = - \sum_{\ell} \sum_{\langle i,j \rangle_\ell} J_\ell(r_i,r_j) s_i s_j - \sum_{\ell<\ell'} \sum_{(i,j) \in C_{\ell,\ell'}} K_{\ell,\ell'} s_i s_j - \sum_{i} h_i(t) s_i,
\end{equation}
where $J_\ell$ are intra-level couplings (rule-dependent), $K_{\ell,\ell'}$ are inter-level couplings, and $h_i(t)$ are external fields.

\subsection{Phase Transitions and Critical Behavior}
The system exhibits second-order phase transitions characterized by order parameters $m_\ell = \langle s_i \rangle_\ell$ measuring coordination within each level. Critical temperature $T_c$ is determined by the spectral condition $\lambda_{\max}(M) = 1$ where
\begin{equation}
    M_{\ell\ell'} = \beta J_{\text{eff}}^{(\ell)} \delta_{\ell\ell'} + \beta K_{\text{eff}}^{(\ell,\ell')} (1-\delta_{\ell\ell'}).
\end{equation}

Near criticality, magnetization scales as $\langle|m_\ell|\rangle \sim (T_c - T)^{\beta_\ell}$ and susceptibility as $\chi_\ell \sim |T - T_c|^{-\gamma_\ell}$, with exponents $(\beta_\ell, \gamma_\ell)$ characterizing universality classes.

\begin{table}[ht]
    \centering
    \caption{Hierarchical cooperation universality classes}
    \label{tab:universality}
    \small
    \begin{tabular}{llll}
        \toprule
        Class & Coupling Regime & Critical Exponents & Coordination Mode \\
        \midrule
        I & Independent ($\kappa \to 0$) & $\beta \approx 0.33$, $\gamma \approx 1.24$ & Layer autonomy \\
        II & Weak inter-level ($\kappa \ll u$) & Perturbative corrections & Matrix organization \\
        III & Strong inter-level ($\kappa \sim u$) & $\beta = 0.5$, $\gamma = 1.0$ & Centralized control \\
        IV & Hierarchical mixed & $\beta = \beta_0 + c_1/L$ & Multi-scale coupling \\
        \bottomrule
    \end{tabular}
\end{table}

\textbf{Class I (Independent layers):} When inter-level coupling $K_{\ell,\ell'} \to 0$, each level transitions independently with 3D Ising exponents ($\beta \approx 0.326$, $\gamma \approx 1.237$, $\nu \approx 0.630$), appropriate for siloed teams with minimal coordination.

\textbf{Class II (Weak coupling):} Perturbative regime $0 < K_{\ell,\ell'} \ll J_\ell$ exhibits corrections to single-layer behavior. Cross-level correlations decay as $\langle \phi_\ell(x) \phi_{\ell'}(x') \rangle \sim |x - x'|^{-2(d-2+\eta)} |{\ell} - {\ell'}|^{-\Delta_{\ell\ell'}}$ with anomalous dimension $\Delta_{\ell\ell'}$.

\textbf{Class III (Strong coupling):} When $K_{\ell,\ell'} \sim J_\ell$, effective dimensionality increases to $d_{\text{eff}} = d + (L-1)$. For $d_{\text{eff}} > 4$ (upper critical dimension), mean-field exponents emerge ($\beta = 0.5$, $\gamma = 1.0$, $\nu = 0.5$), characteristic of tightly coupled hierarchies with strong top-down control.

\textbf{Class IV (Hierarchical):} Novel fixed point at $K_{\ell,\ell'} = J_\ell \sqrt{L}$ yields logarithmic corrections: $\beta_H = \beta_0 + c_1/L + O(L^{-2})$ and $\nu_H = \nu_0(1 + c_2 \ln L)$, with universal constants $\{c_1, c_2\}$ characterizing hierarchical structure.

\subsection{Early Warning Indicators}

Critical slowing down near phase transitions manifests through four complementary signals that enable predictive intervention before coordination breakdown:

\textbf{1. Autocorrelation timescale.} The characteristic relaxation time diverges as $\tau_{\text{AC}} \sim |T - T_c|^{-\nu z}$ where $z$ is the dynamic critical exponent. Warning threshold: $\tau_{\text{AC}}/\tau_{\text{baseline}} > 5$.

\textbf{2. Variance amplification.} Fluctuations grow via $\text{Var}[\Phi_\ell] \sim |T - T_c|^{-\gamma}$ as the system becomes more susceptible. Warning threshold: $\text{Var}[\Phi_\ell]/\text{Var}_{\text{baseline}} > 3$.

\textbf{3. Transfer entropy lag.} Information propagation delays scale as $\tau^* \sim |T - T_c|^{-\nu z}$. Warning threshold: $\tau^* > 2\tau_{\text{actuation}}$.

\textbf{4. Sobol sensitivity explosion.} Variance-based sensitivity indices diverge as $S_i \sim |T - T_c|^\gamma$. Warning threshold: $\max_i S_i > 0.7$.

We combine these into a composite early warning index:
\begin{equation}
    \mathcal{W}_{\text{EWS}} = w_1 \log(\tau_{\text{AC}}) + w_2 \text{FDV}_{\text{avg}} + w_3 \frac{\diff}{\diff t}\langle D_{\text{KL}}\rangle + w_4 \Delta\text{TE}^{\text{reversal}},
    \label{eq:early-warning}
\end{equation}
where FDV measures fluctuation-dissipation violations and $\Delta\text{TE}^{\text{reversal}}$ detects directional inversions in information flow. Alert levels trigger at $\mathcal{W}_{\text{EWS}} > 3\sigma$ (warning) and $\mathcal{W}_{\text{EWS}} > 5\sigma$ (critical), enabling proactive governance adjustments or temperature modifications before performance degradation.

\section{Coordination Mechanisms}
We present four design levers that consistently improved performance across scenarios, grounded in multi-agent learning theory and information-theoretic analysis.

\subsection{Consensus with Safety Constraints}
Distributed coordination uses a stochastic matrix $C \in \mathbb{R}^{n \times n}$ encoding influence weights $c_{ij} \geq 0$ with row sums equal to one. Convergence speed is characterized by spectral radius $\rho(C)$.

\begin{proposition}[Safe Consensus]
If $C$ is doubly stochastic with $\rho(C - (1/n)\mathbf{1}\mathbf{1}^\top) < 1$, then the update $x_{t+1} = \Pi_S(Cx_t)$ converges to consensus within safe set $S$ when $S$ is convex and closed.
\end{proposition}

This projection-based approach ensures governance constraints remain satisfied throughout coordination, combining efficiency with safety guarantees.

\subsection{Thermodynamically Governed Noise Injection}
Noise injection follows Langevin dynamics with Fokker-Planck evolution, ensuring thermodynamic consistency via the fluctuation-dissipation theorem. Agent dynamics obey
\begin{equation}
    \frac{\diff x_i}{\diff t} = -\mu \nabla U_\ell(x_i) + \sqrt{2\mu k_B T_\ell}\, \xi_i(t),
    \label{eq:langevin}
\end{equation}
where $U_\ell$ combines value functions and governance penalties, $\mu$ is mobility, and $\xi_i(t)$ is white noise. The Einstein relation $D_\ell = \mu k_B T_\ell$ connects diffusion coefficient to temperature.

Order parameter variance evolves as
\begin{equation}
    \frac{\diff \sigma_\ell^2}{\diff t} = -2\mu k_\ell \sigma_\ell^2 + 2\mu k_B T_\ell d,
    \label{eq:variance-dynamics}
\end{equation}
equilibrating at $\sigma_{\text{eq}}^2 = k_B T_\ell d / k_\ell$. We employ variance-based adaptive cooling:
\begin{equation}
    T_\ell(t+1) = \begin{cases}
        \alpha T_\ell(t) & \text{if } \rho(t) \in [0.9, 1.1] \text{ (equilibrated)} \\
        T_\ell(t) & \text{otherwise (still exploring)}
    \end{cases}
    \label{eq:adaptive-cooling}
\end{equation}
where $\rho(t) = \sigma_\ell^2(t)/\sigma_{\text{eq}}^2(T_\ell)$ measures relaxation progress.

Regime switching follows Kramers escape rates $k_{A \to B} = (\omega_A \omega_B)/(2\pi\xi) \exp(-\Delta U_\ell/(k_B T_\ell))$, with mean residence time $\tau_A \sim \exp(\Delta U_\ell/(k_B T_\ell))$. Temperature initialization uses $T_\ell(0) = \Delta U_\ell/(k_B \ln H)$ for planning horizon $H$. Level-dependent stratification matches timescales: $T_\ell/T_{\ell+1} = \tau_{\ell+1}/\tau_\ell$, maintaining higher temperatures at faster lower levels.

\subsection{Hierarchical Bandit Learning}
Tool and option selection follows a contextual bandit framework with linear reward model $\mathbb{E}[r_t \mid c_t, a_t] = c_t^\top \theta_{a_t}$. Under hierarchical feedback with delay $d_\ell$, LinUCB achieves regret bound
\begin{equation}
    R_T = O\bigl(d\sqrt{T \log(1 + T/\lambda)} + d_\ell\bigr),
\end{equation}
where $d$ is context dimension. This quantifies the learning efficiency cost of hierarchical coordination.

\subsection{Information Flow and Thermodynamic Coupling}
Transfer entropy $TE_{\ell \rightarrow \ell+1}$ measures directional causal influence between hierarchical levels \cite{hoel2013}. A fundamental bound connects information flow to thermodynamic cost:
\begin{proposition}[Information-Thermodynamic Duality]
Transfer entropy is upper-bounded by driven entropy production:
\begin{equation}
    TE_{\ell \rightarrow \ell+1} \leq \sigma_{\ell+1}^{\text{driven}},
    \label{eq:te-entropy-bound}
\end{equation}
where $\sigma_{\ell+1}^{\text{driven}} = k_B \langle J \cdot \ln(\pi w / \pi' w') \rangle$ quantifies irreversible dissipation from coordination.
\end{proposition}

This bound implies that low transfer entropy with high entropy production indicates inefficient coordination—energy is dissipated without effective information transmission. Conversely, high $TE$ with low $\sigma$ signifies near-reversible coordination where supervisory directives align with natural dynamics.

Aggregation efficiency $\eta_{\ell \rightarrow \ell+1} = I(S_\ell; S_{\ell+1})/H(S_\ell)$ quantifies information preservation during upward aggregation, with $\eta = 1$ achieved for injective mappings. We monitor the KL divergence cascade $D_{\text{total}} = \sum_\ell \alpha_\ell D_{\text{KL}}(p_\ell \| \pi_\ell^*)$ from target distributions $\pi_\ell^*$, with propagation inequality $D_{\text{KL}}(p_{\ell+1} \| \pi_{\ell+1}^*) \leq C_\ell D_{\text{KL}}(p_\ell \| \pi_\ell^*) + \epsilon_{\text{agg}}$ establishing quantitative coordination targets.

A drop in $TE_{\ell \rightarrow \ell+1}$ coupled with rising constraint violations or increasing $D_{\text{KL}}$ signals that summaries fail to convey actionable state, prompting schema refinement or aggregation operator redesign.

